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Properties

Label 8-1152e4-1.1-c3e4-0-4
Degree 88
Conductor 17612.050×10817612.050\times 10^{8}
Sign 11
Analytic cond. 2.13439×1072.13439\times 10^{7}
Root an. cond. 8.244408.24440
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·17-s + 428·25-s − 1.24e3·41-s − 476·49-s − 984·73-s − 5.56e3·89-s − 1.20e3·97-s − 6.20e3·113-s + 1.74e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.98e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 0.456·17-s + 3.42·25-s − 4.75·41-s − 1.38·49-s − 1.57·73-s − 6.63·89-s − 1.26·97-s − 5.16·113-s + 1.30·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.63·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯

Functional equation

Λ(s)=((228⋅38)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((228⋅38)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 228⋅382^{28} \cdot 3^{8}
Sign: 11
Analytic conductor: 2.13439×1072.13439\times 10^{7}
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 228⋅38, ( :3/2,3/2,3/2,3/2), 1)(8,\ 2^{28} \cdot 3^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 1.2602749511.260274951
L(12)L(\frac12) ≈\approx 1.2602749511.260274951
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
good5C22C_2^2 (1−214T2+p6T4)2 ( 1 - 214 T^{2} + p^{6} T^{4} )^{2}
7C22C_2^2 (1+34pT2+p6T4)2 ( 1 + 34 p T^{2} + p^{6} T^{4} )^{2}
11C22C_2^2 (1−870T2+p6T4)2 ( 1 - 870 T^{2} + p^{6} T^{4} )^{2}
13C22C_2^2 (1−3994T2+p6T4)2 ( 1 - 3994 T^{2} + p^{6} T^{4} )^{2}
17C2C_2 (1−8T+p3T2)4 ( 1 - 8 T + p^{3} T^{2} )^{4}
19C22C_2^2 (1−6550T2+p6T4)2 ( 1 - 6550 T^{2} + p^{6} T^{4} )^{2}
23C22C_2^2 (1−4338T2+p6T4)2 ( 1 - 4338 T^{2} + p^{6} T^{4} )^{2}
29C22C_2^2 (1−46662T2+p6T4)2 ( 1 - 46662 T^{2} + p^{6} T^{4} )^{2}
31C22C_2^2 (1+59134T2+p6T4)2 ( 1 + 59134 T^{2} + p^{6} T^{4} )^{2}
37C22C_2^2 (1−74410T2+p6T4)2 ( 1 - 74410 T^{2} + p^{6} T^{4} )^{2}
41C2C_2 (1+312T+p3T2)4 ( 1 + 312 T + p^{3} T^{2} )^{4}
43C22C_2^2 (1+20186T2+p6T4)2 ( 1 + 20186 T^{2} + p^{6} T^{4} )^{2}
47C22C_2^2 (1+178974T2+p6T4)2 ( 1 + 178974 T^{2} + p^{6} T^{4} )^{2}
53C22C_2^2 (1−226998T2+p6T4)2 ( 1 - 226998 T^{2} + p^{6} T^{4} )^{2}
59C22C_2^2 (1−346246T2+p6T4)2 ( 1 - 346246 T^{2} + p^{6} T^{4} )^{2}
61C22C_2^2 (1−436538T2+p6T4)2 ( 1 - 436538 T^{2} + p^{6} T^{4} )^{2}
67C22C_2^2 (1−343478T2+p6T4)2 ( 1 - 343478 T^{2} + p^{6} T^{4} )^{2}
71C22C_2^2 (1+257070T2+p6T4)2 ( 1 + 257070 T^{2} + p^{6} T^{4} )^{2}
73C2C_2 (1+246T+p3T2)4 ( 1 + 246 T + p^{3} T^{2} )^{4}
79C22C_2^2 (1+931870T2+p6T4)2 ( 1 + 931870 T^{2} + p^{6} T^{4} )^{2}
83C22C_2^2 (1−195606T2+p6T4)2 ( 1 - 195606 T^{2} + p^{6} T^{4} )^{2}
89C2C_2 (1+1392T+p3T2)4 ( 1 + 1392 T + p^{3} T^{2} )^{4}
97C2C_2 (1+302T+p3T2)4 ( 1 + 302 T + p^{3} T^{2} )^{4}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.70452409333778366591594433586, −6.53481314099422948401845256461, −6.49269611964710388689063653938, −5.65347616855220775776091121568, −5.59942407148581592771416956372, −5.56590742878226441953638798846, −5.40490202981920661751287285999, −4.88867339007516906535949521292, −4.85134172251848842998253374235, −4.55380884912313213330433087619, −4.54520414980913956935308001048, −3.91959040054945740303870633145, −3.77781344438274703600685602465, −3.67275405902544004804967415232, −3.11875767581333603192759740542, −2.96907258457680943611021892785, −2.82864035433359919397687291548, −2.70417249903300656061131754117, −2.24457335923841451895262156729, −1.51086005701955343445194400258, −1.50701856831571009167156852478, −1.45568381472207773888265977932, −1.10105881699621432751345220702, −0.39322742959378937911642672746, −0.17824465339069045753305772108, 0.17824465339069045753305772108, 0.39322742959378937911642672746, 1.10105881699621432751345220702, 1.45568381472207773888265977932, 1.50701856831571009167156852478, 1.51086005701955343445194400258, 2.24457335923841451895262156729, 2.70417249903300656061131754117, 2.82864035433359919397687291548, 2.96907258457680943611021892785, 3.11875767581333603192759740542, 3.67275405902544004804967415232, 3.77781344438274703600685602465, 3.91959040054945740303870633145, 4.54520414980913956935308001048, 4.55380884912313213330433087619, 4.85134172251848842998253374235, 4.88867339007516906535949521292, 5.40490202981920661751287285999, 5.56590742878226441953638798846, 5.59942407148581592771416956372, 5.65347616855220775776091121568, 6.49269611964710388689063653938, 6.53481314099422948401845256461, 6.70452409333778366591594433586

Graph of the ZZ-function along the critical line